See e.g., [1]}. Applying [2]} to this Hopf \(*\) -algebra with its finite dimensional type 1 \(*\) -representations, we obtain the compact quantum group \(\mathrm {U}_q(n)\)  with Hopf \(*\) -algebra \((\mathbb {C}[\mathrm {U}_q(n)],\Delta _n,S_n,\varepsilon _n)\) , which is exactly the coordinate ring \(A(\mathrm {U}_q(n))\)  of the so-called quantum unitary group of rank \(n\)  discussed in e.g. [3]},[4]} (see [5]} for its concise review) in the context of \(q\) -analysis. (Note that the choice of generators of \(U_q\mathfrak {gl}(n)\)  here follows [3]} (rather than [4]}) with \(K_i^{\pm 1} := q^{\pm \epsilon _i}, x_j := e_j, y_j := f_j\) .)
